Time Accurate Solutions of the Fokker-Planck Equation in Simple Shear

Existing efforts in simple shear flow have deemed solving the Fokker-Planck (FP) equation for fiber orientation kinetics using the centered differencing scheme (CDS) with an explicit time integration scheme (ETS) unstable, expensive, and inaccurate at high Peclet numbers (Pe). This work shows that conservative solutions may be obtained for the FP equation using this combination of schemes at a low computational cost. The FP equation is solved on unstructured cubed-sphere grids using the finite-volume method. The ETS is a two-stage second-order Runge-Kutta scheme. A modified Jeffery equation (Ferec et al. Rheol. Acta (2014) 53:445–456) is employed to solve for the time evolution of the orientation vector. The approach accounts for variable shape factor and rotational diffusion coefficient. Because an ETS is employed, the coupling of the fiber orientation probability density function with the shape factor and rotational diffusion coefficient does not require linearization. The solver is rigorously tested to show that the CDS does not require stabilization in the Pe interval considered. Solutions are obtained for non-dilute and semi concentrated suspensions up to Pe=100000. We will demonstrate that the solver aids the development of constitutive relations and closure models.